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JOURNAL OF ALGEBRA I@473488 (1986)

Finite Resolutions of Modules for Reductive Algebraic Groups

S. DONKIN

School of Mathematical Sciences, Queen Mary College, Mile End Road, London El 4NS, England

Communicafed by D. A. Buchsbaum

Received January 22, 1985

INTRODUCTION

A popular theme in recent work of Akin and Buchsbaum is the construc- tion of a finite left resolution

of a suitable G&-module M by modules which are required to be direct sums of tensor products of exterior powers of the natural representation. In particular, in [Z, 31, for partitions 1 and p, resolutions are constructed for the modules L,(F) and L,,,(F) (the Schur functor corresponding to 1 and the skew Schur functor corresponding to (A, p), evaluated at F), where F is a free module over a or the . The purpose of this paper is to characterise the GL,-modules which admit such a resolution as those modules which have a filtration in which each successivequotient has the form L,(F) for some partition p. By contrast with [2,3] we do not produce resolutions explicitly. Our methods are independent of those of Akin and Buchsbaum (we give a new proof of their result on the existence of a resolution for L,(F)) and are algebraic theoretic in nature. We have therefore cast our main result (the theorem of Section 1) as a statement about resolutions for reductive algebraic groups over an algebraically closed field. One may frequently wish to work over a more general coefficient (see [l-3]) and so we treat in some detail Chevalley group schemes and general schemes over a principal ideal domain (2.6 and 2.7), giving the appropriate versions of our main result and relating it to the work of Akin, Buchsbaum and Weyman.

473 0021-8693/86 $3.00 Copyright ?: 1986 by Academic Press, Inc. All rights of reproduclmn m any form reserved. 474 S. DoNKIN

1. RESOLUTIONSAND FILTRATIONS

Let G be a reductive, connected, linear over an algebraically closed field K. Let T be a maximal torus of G and let @ be the root system of G with respect to T. Let B denote the Bore1 of G containing T such that the non-zero weights of the adjoint action of T on the Lie(B) of B are the negative roots. Let X denote the charac- ter group of T. There is a natural partial on X: we write j” < p if p - 3. is a sum of positive roots. All modules considered here for an algebraic group over K will be rational (see Section 1 of [ 131) and finite dimensional over K. For 1,E X we denote by K, the one dimensional B-module on which T acts with weight %. The set X+ of dominant weights consists of those k E X such that the induced module (see [ 111) Kj, 1g is non-zero and we write Y(n) for KL 1g. For I dominant Y(n) has a unique highest weight I which occurs with multiplicity one (see, for example, (1.4.3) of [ 141). A filtration of a G-module is declared to be good if each successivequotient is either 0 or isomorphic to Y(1) for some 1 E X+. We denote by 6 the class of G-modules which have a good filtration and by 6* the class of G- modules V such that the dual module V* = Hom,( V, K) belongs to 6. (If G is semisimple and simply connected then VE 8* means that V has a Weyl filtration; see [20, 5.23.) For VE 8 the multiplicity of Y(J) as a successive quotient in a good filtration is, by (12.1.l ) of [ 141, independent of which good filtration is chosen and we denote this multiplicity by (V: Y(n)).

DEFINITION. A special resolving system is a subclass 6 of 6 n 6* which is closed under the formation of direct sums and tensor products and which contains, for each 1 E X+, some module CA such that Cj. has unique highest weight i and ,I occurs with multiplicity one. Let B denote a special resolving system. We shall need the following sim- ple fact.

LEMMA 1. Zf 0 + V’ + V + V” + 0 is a short exact sequence of G- modules with V, V’ and V” in 8 and CE (5 then

0 -+ Horn&C, V’) -+ Hom,( C, V) + Hom,( C, V”) -+ 0

is exact. ProojI By the long exact sequence of cohomology it is enough to prove that the Hochschild cohomology group Ext&(C, V’) is 0. Now Extk(C, v’) rH’(G, C*@ v’) and, since C*, V’E 6, C* @IV’ has a filtration with successive quotients of the form Y(A)@ Y(p), I”, p E X+. RESOLUTIONS OF MODULES FOR REDUCTIVE GROUPS 475

However, it is a result of Cline, Parshall, Scott and van der Kallen ((3.3) Corollary of [ 133 for semisimple groups; see also (2.1.4) of [ 143 for reduc- tive groups) that H’(G, Y(n)@ Y(p)) =0 for all i>O. Hence H’(G, C* @ I”) = 0 and the proof is complete. Our main result is that a special resolving system is a resolving system for 8, or more precisely:

THEOREM. A G-module M has a finite left 6 resolution if and only if ME6 If M has a finite left (5 resolution then there is an exact sequence o-+ v,-+ v+, + . . . + V, + V, + M + 0 with all Vi E (li. It follows from Proposition 3.2.4 of [14] and induction on m (see also 4.1 (2) of [6]) that M has a good filtration. To establish the reverse implication we use induction on the weights of M. For a G-module S we set o(S) = { ,u E X+: p is a weight of S}. The for- mal character of Y(n) (A E X’ ) is given by Weyl’s character formula (see, e.g., [4, (2.2.7)]) so it follows, from 21.3 Proposition of [7], that o( Y(A)) is saturated in the sense that whenever r E o( Y(n)), p E X+ and p < r, then ALEo( Y(n)). Thus w(N) is saturated for any G-module N with a good filtration.

LEMMA 2. For any non-zero ME 6 there exists a short exact sequence of G-modules

O+D’-+D-+M+O with DE 6, D’ E 6 and such that o(D’) is a proper subset of o(M). The Theorem follows now from Lemma 2 by induction on lo(M)1 by combining a resolution 0 + C, -+ C, _ i -+ . . + C,, + D’ + 0 with the short exact sequence 0 -+ D’ + D + M+O to form a resolution o+c,+ ... +C,,+D-*M+O ofM. Proof of Lemma 2. We prove Lemma 2 by induction on lo(M Assume the result to be true for all non-zero NE@ with lo(N)1 < lo(M Let 1 be a highest weight of M. By Lemma 3.2.3 of [14] there is a sub- module M’ of M such that M’ E 8, M/M’ is a of copies of Y(n) (say n of them) and 1$ u(M’). Also by Lemma 3.2.3 of [ 141 there is a sub- module, say CA, of Cl such that C; E 8,1$ o(C;) and C,/C, is isomorphic to Y(1). Thus we may choose an epimorphism rc: C= @:= 1 CA+ M/M with C’ = @;= 1 C”. By Lemma 1, there is a morphism cp:C + M such that qcp= rc, where q: M+ M/M’ is the quotient map. Let $: C@ M’ + M be the sum of cpand inclusion. Then II/ is surjective and the kernel consists of those pairs (c, m’) E CO M’ with q(c) + m’ = 0, i.e., the 476 S. DONKIN pairs (c, --q(c)) with cp(c)~M’. Now q(c) belongs to M’ if and only if qcp(c)= 0, i.e., if and only if X(C)= 0, that is, if and only if c E c’. Hence the kernel of $, say E, is {(c, -q(c)): c E Cl}, which is isomorphic to C’. We have o(M’)_cw(M) and i $u(M’) so that Jw(M’)) < Iw(M)I. If M’ = 0 we may take D = C, D’ = C’ in the statement of Lemma 2. Thus we may assume that M’ # 0 and so, by the inductive hypothesis, there is an epimorphism 5: Q -+ M’, for some Q E 6 with kernel Q’ E 6 satisfying u(Q’) c o(M’). Let [: D = C@ Q + C@ M’ be the sum of the identity map and < and let a: D + M be the composite t+G0 5. We must show that the kernel D’ of a belongs to 6 and that u(D’) is a proper subset of w(M). We have D’ = Ker a = Ker(rj 0i) = [-‘(Ker $) = [ ~ ‘E. The kernel of the restriction of {: D’ -+ E is the kernel of [, i.e., Q’, and the is E. Both Q’ and E (isomorphic to C’) have a good filtration and so D’ E 8. Also, we have

w(D’) = w(Q’) u o(E) = co(Q’) u w(C’) c o(M) u w(C’) since E= c’ and w(Q’) &w(M’). Also, o(M’)cu(M)\{I}, so to complete the proof it suffices to show that o(C’) G o(M)\(J). Now c’ is a direct sum of copies of C>.so that o(C’) = o(C;). Let r be a dominant weight of C;.. Then T is a weight of CI and, since 1 is the unique maximal weight of Cj,, r d II. As 2 is a weight of M and o(M) is saturated we obtain that r belongs to o(M). However, as remarked earlier, 14 w( CL) so that each dominant weight of C’ belongs to w(M)\{ A} and o( C’) G o(M)\{ L}. Thus o(D’) is a proper subset of o(M) and we have a short exact sequence 0 + D’ -+ D -+ M -+ 0 fulfilling the requirements of the lemma. The Theorem also has a dual reading involving right resolutions of modules M such that M* E 6. We leave the precise formulation to the reader.

2. COMPLEMENTS

This section consists of observations on the resolutions produced by means of Section 1, the existence of special resolving systems and the validity of the above theorem for Chevalley groups schemes and schemesover a principal ideal domain.

2.1. The length m of a resolution o-tc, -‘Cm-l+ ... -+C,+M-bO produced by means of Section 1 satisfies m < lo(M)1 - 1. Moreover, the proof of Lemma 2 shows that one may arrange that w(C,) E o(M) for all OGibm. RESOLUTIONS OF MODULES FOR REDUCTIVE GROUPS 477

2.2. Let G be a semisimple, simply connected algebraic group over an algebraically closed field K and let A,,..., II be the fundamental dominant weights. Suppose we are given a special resolving system 6 and let CA, have highest weight & (1~ i< I). We may form the subsystem generated by the CA,, i.e., the class of direct sums of tensor products of the CA,. This will still be a special resolving system since any dominant weight 1 has the form m,A,+ *** + m,lZ, for non-negative integers m,,..., m, and (@“‘C,,) @ *.. @ (@““Cq) has highest weight A. We will call a special resolving system generated in this way by C,,‘s (one for each i between 1 and I) a reduced special resolving system (or simply a reduced system) and the generating C,,‘s will be called a base. Conversely suppose we are given a module CA,E 6 n 6* with unique highest weight Ai, for each 1 < i < 1. It is shown in [14] (see also [22]) that the class of modules, for a connected algebraic group over an algebraically closed field K, which admit a good filtration is closed with respect to the formation of tensor products, provided that either the of K is not 2 or the group does not involve E, or Es. Thus the direct sums of ten- sor products of the CA, (1 < i < 1) form a reduced special resolving system (except possibly in the bad casesjust mentioned). 2.3. For G = 2X,(K), with K an algebraically closed field, there is a uni- que reduced system. This is because for i a fundamental dominant weight there is no VEX+ with PC I (i.e., I is a miniscule weight; see [S, p. 1271) so that C1 must be isomorphic to Y(1). Thus any reduced system is generated by the Y(n) for 1 fundamental. Moreover Y(n) for 1 fundamental is isomorphic to /i’V, for some 1 < r < n, where V is the natural module for G. Thus the only candidate for a reduced system is simply the restriction to Z,(K) of the system for G&(K) considered by Akin and Buchsbaum [2]. Of course we should point out that the nrI’ do generate a special resolving system. By 2.2 we need only check that I’VE 6* and this is true since the dual of nrV (1

481/101/2-13 478 S. DONKIN teristic p of K satisfies p 3 (,li + p, &) for all 1 6 i < 1, where E,,,..., IV1are the fundamental dominant weights and /?I;= 2&/(/I,,, PO).Then each belongs to the bottom alcove of X+ - p and so each Y(~i) is irreducible (see Section 1.8 of [ 181). It follows that the dual module Y(&)* is irreducible of highest weight -w&, where wO is the longest element of the Weyl group. However, - wOlzi is a fundamental weight also and so is equal to A, for some 1 (Ai + p, &) for all 1 d id 1 is p 2 35. 2.5. We now consider the possibility of defining a special resolving system by exterior powers. Let V be a finite dimensional vector space of dimension n over an algebraically closed field K of characteristic p. For G = Sp( V) (n even), A’I’ has a good filtration for i < n by Section 4.4 of [6]. The fundamental weights 1, ,..., 1, may be arranged so that /1’V has highest weight 1,. Also (/iV)* z.~‘V* gAiV so that, by 2.2, the ,4’V, 1~ i 7 is sufficient). We omit the details. Also, we leave open the problem of describing a special resolving system for the groups of type B, and D, (and indeed for the exceptional groups) in characteristic 2 and the related problem of finding special resolving systems for the corresponding Chevalley group schemesover Z (see2.6). 2.6. It is our intention in this subsection to convince the reader that the analogue of the Theorem of Section 1 is valid for Chevalley group schemes over a principal ideal domain; in 2.7 we treat the general linear group schemes.What we shall do, in effect, is to check that the results for reduc- tive groups, which we have used in the proof of the Theorem, are also valid for Chevalley group schemes. Let R be a principal ideal domain and G an affine group over R whose coordinate ring R[_G] is flat over R. An account of the represen- tation theory of such a is to be found in Section 1 of [S], to which we frequently refer. By a rational (left) G-module we mean a right R[G]-comodule. A rational G-module I’, with structure map z: V+ V Q R R[_G], may also be regarded as a module (in the usual sense) for the group ring R_G(R). The action of 1;(R) = Hom,+,,(R[_G], R) on V is given by gu = (1@g) z(u), for g E G(R), u E V, where 1: V + V is the iden- RESOLUTIONS OF MODULES FOR REDUCTIVE GROUPS 479 tity map. If R is an algebraically closed field and R[_G] is a finitely generated, reduced R-algebra then this action determines an equivalence of categories between rational G-modules and G(R)-modules which are rational in the usual sense (see [12, Sect. 11). The of rational G- modules is an abelian category with enough injectives ((1.2) and (1.5) of [S] ) so that we have available the derived functors of various left exact functors. We now specialize to the case of Chevalley group schemes,The details of the construction and properties of these are to be found in [ 10, 7,211. Let g be a finite dimensional, complex, and rc:g + g1(V) a representation of g such that the weights of V span the of integral weights of a Cartan subalgebra of g. There is defined, for each field k, a Chevalley group Gn,k (depending on the choice of a Chevalley basis of g and an admissible Z-form of V), We have the Chevalley Z-form Z[G,] of the coordinate ring of the complex Chevalley group G,c. The Chevalley Z- form is the coordinate ring of an afine group scheme, G say, over Z and so we are entitled to write Z[ G,) = Z[G]. The construction of Z[G,] depends on the choice of a Chevalley basis of g and associated to this choice there is a distinguished maximal torus _T,negative Bore1 subgroup B and maximal subgroup _Uc &. For each integral weight A we have a rational &module ZA, free of rank one as a Z-module, on which _Uacts trivially and _Tacts with weight A. Let R be a principal ideal domain. We obtain, by base extension, group schemes GR, BR, etc., and a RR-module RA = R Q z ZA for each integral weight A. Of crucial importance in the proof of the Theorem of Section 1 for a Chevalley group scheme is (3) below, and the proof of (3) rests on the following. (1) (Kempfs Vanishing Theorem) For A dominant, Hi(GR/JIR, R,) = 0 for all i > 0. Here, as in [S], we denote by Hi(_GR/BR, -) the derived functors of the induction functor from B,-modules to &-modules. For an algebraically closed field K, Hi(GK/_BK, KJ = 0 for all i > 0 and 1 dominant by Kempf s Vanishing Theorem in its usual formulation (see, e.g., [4]). Thus we obtain from the Universal Coefficients Theorem, l.l8(ii) of [S], that for any ring homomorphism from R into an algebraically closed field K, Hi(&/BR, R,) ORK= 0. But Hi(L;J_BR, R,J if a finitely generated R- module (Remark 1.21 of [S] ) and consequently must be 0. Suppose we have a ring homomorphism R + R’ into a principal ideal domain R’. From the Universal Coefficients Theorem, l.l8(ii) of [S] with i = 0 we now obtain from (1): (2) For 1 dominant, the natural map R’ OR Y,(A) --t YRS(IZ) is an isomorphism of _G,,-modules. 480 S. DONKIN

Here we denote by Y,(J) (resp. Y,.(i)) the induced module fW,/B,, RJ (rev. fWR81BRs, R>.)). Any rational &module which is finitely generated as an R-module is the direct sum of its weight spacesand one defines for a _T,-module, which is finitely generated and free over R, its formal character (see [S, p. 5021). By virtue of (1) above and Corollary 2.7(ii) of [S], the character of Y,(A), for 2 dominant, is given by Weyl’s character formula. In particular Y,(n) has unique highest weight A and one may apply the argument of Cline, Parshall, Scott and van der Kallen, (3.3) of [13], to obtain: (3) For dominant weights A and u, H’(_GR, Y,(n)@ Y&t)) = 0 for all i> 0. We leave it to the reader to formulate the notions of special resolving systems and good filtrations for G,-modules. Let 6 be a special resolving system. The correct analogue of the Theorem of Section 1 for & is: (4) Let M be a rational &-module which is finitely generated over R. Then M has a good filtration tf and only tf M is torsion free over R and has a finite left Gresolution. The implication that a module M with a good filtration has a finite left t&resolution may be obtained by use of weight spaces, (3) above and the arguments of Section 1. It remains to prove the reverse implication, which amounts to showing: (5) ZfO+ v,+ vnpl* ... + V, + M -+ 0 is an exact sequence of &- modules, finitely generated over R, such that all the Vi have good filtrations and M is torsion free over R, then A4 has a good filtration. If R is an algebraically closed field then (5) is true by induction on n and Proposition 3.2.4 of [14] (cf. 4.1, (2) of [6]). To prove (5) for an arbitrary principal ideal domain R we develop a useful criterion for a &-module to have a good filtration. (6) A rational S;,-module M, which is finitely generated over R, has a good filtration if and only tf M is free over R and for every ring homomorphism from R into an algebraically closed field K the _G=module M, = K 0 R M has a good filtration. Note that (5) follows easily from (6). Since each term in the exact sequenceof (5) is free over R, tensoring with K gives an exact sequence of Grmodules. Each K OR Vi has a good filtration, as a &-module, by (2) SO that M, has a filtration by the remark following (5). Thus by (6), M has a good filtration. A version of (6) for modules for the Kostant Z-form U, of the envelop- ing algebra of g occurs in [14, (11.5.3)] and it is possible to deduce (6) RESOLUTIONS OF MODULES FOR REDUCTIVE GROUPS 481 from a modified form of this by setting up a correspondence between UR = R @ ,U,-modules and _G,-modules. However, we prefer to give a proof of (6) more in keeping with the spirit of this paper. If M has a good filtration then so does M, by (2). Now suppose that MK has a good filtration for each algebraically closed field K. We proceed by induction on the rank of M. Suppose that the rank of M is greater than 0 and the result holds for modules of smaller rank. Let A be a maximal weight of M occurring with multiplicity n, say. Then there is a B,-module epimorphism cp:A4 + Rp), the direct sum of n copies of R,. By the univer- sal property of induction [S, (1.7)], there is a unique _G,-module map 6: A4 + Y,(1)‘“’ such that so 8 = cp, where E: Y,(A)@) + RI;) is evaluation. Thus we obtain, for an algebraically closed field K, a _G=module map 0,: M, -+ K@ Y,(A)cfl) which restricts to an isomorphism on 1 weight spaces. By Lemma 3.2.3 of [ 141 there is a submodule, say Z(K), of M, such that 1 is not a weight of Z(K), Z(K) has a good filtration, and M,/Z(K) is a direct sum of n copies of YK(Iz). Since I, and hence wJ. (where w,, is the longest element of the Weyl group), is not a weight of Z(K) we must have B,(Z(K)) = 0, by [14, (1.5.2)]. Thus we have Z(K) < Ker 8,. However, dim,(M,/Z(K)) = n. dim, Y,(A) and dim,(M,/Ker 0,) = dim, Im eK < n. dim, Y,(A) (by (2)) so that we must have Z(K) = Ker 0, and 0, is surjective. We now have that 8 is a map between finitely generated R-modules such that, for each morphism R + K into an algebraically closed field, the extension 6, is surjective. Hence 0 is surjective. Let Z be the kernel of 6’ and identify ZK with a &-submodule of M,, for K an algebraically closed field. By dimensions we have Z, = Z(K) so that Z, has a good filtration. By the inductive hypothesis, Z has a good filtration. We have now shown that Z and M/Z have good filtrations and so A4 has a good filtration. The proof of (6) is now complete. We have already made use of (6) in proving (5), and hence (4). We now give further applications. (7) Suppose that either 8 involves no component of type E, or E,, or that 2 is a unit in R. Then, for every pair of dominant weights A and p, the YR(l) 0 Y&) has a good filtration. This is true if R is an algebraically closed field by [14, (10.8.5)] the Theorem and so is true for an arbitrary principal ideal domain R by (6) above. There remains the problem of existence of special resolving systems for Chevalley group schemes.Let g = s/,(C), let rr: g + gl,(C) be inclusion and let _Gbe the Chevalley group scheme over Z constructed using the natural Chevalley basis (see [9, 11.2.11) and the admissible lattice M, consisting of the column vectors with entries. Then MK is the natural GK = X,(K)-module for K an algebraically closed field. It follows from 2.2, 482 S. DONKIN

(7), 2.3 and (6) that the A’M,, 1 d i < n, form a base of a special resolving system for the Chevalley group scheme_GR over a principal ideal domain R. Similarly, for g = sp(2n, C), one may take /i iM, (1 G id n) as a base for a special resolving system for GR, where M is the natural G-module. For g of type B, or D, one may also construct, as in 2.5, a special resolving system for GR using exterior powers, provided that 2 is a unit in R. 2.7. We now come to the general linear group schemes and relate our material to the work of Akin, Buchsbaum and Weyman [l-3]. We form the free commutative ring B[X, J in n* variables X, (1 6 i6 n, 1

(2) For all 1, p E A+, the _G.-module Y,(A) @I Y,(p) has a goodfiltration. The dependence of (2) on [ 143 is not as serious as that of 2.6, (7)-since RESOLUTIONS OF MODULES FOR REDUCTIVE GROUPS 483

we are only interested in groups of type AI the statement (2) depends essen- tially only on [22, Theorem B]. In [l], the Schur functor evaluated at a free module is defined as the image of a map from a tensor product of exterior powers to a tensor product of symmetric powers. So to relate the above to Schur functors we first consider exterior and symmetric powers. We denote by E,EA (1~ i< n) the n-tuple with 1 in the i-component and 0 elsewhere. Let A4 denote the natural module for C, i.e., the free Z-module on the basis ml, m2,..., m, and comodule map z: M + MO $?[C] satisfying z(m,) = J$= I mj@ Xii (1 Q id n). We obtain, by base extension, a GR- module F = M,.

(3) For 0 d m

El+ . . . + E, and there is a unique (up to isomorphism) irreducible module of highest weight 1 for each 1 EA+. Thus by (6) of 1.6 and (1) above A”‘M, has a good filtration for the arbitrary principal ideal domain R. However, for example, by Weyl’s Character Formula, nrnA4, and Y,(E, + ... +E,) have the same character so a good filtration of AmM, can have only one non-zero quotient and that must be isomorphic to yR(El+ . . . + E,). (4) For any r > 0, the r th symmetric power S,F is isomorphic, as a &-module, to Y,(re,). For R = K-an algebraically closed field-this is well known. One way in which it may be obtained is from the fact that the r-fold tensor product O’F has, by (2) a good filtration and, by Proposition 3.2.6 of [ 141, one may arrange for the term YK(re,) to be at the top of the filtration. Thus there is an epimorphism @‘F + YK(rEl) and this epimorphism induces an epimorphism S,F+ YK(rE,), by (1.5.2) of [14], since r.zl is not a weight of the submodule of @‘F involved in the definition of the r-fold symmetric power. But S,F and YK(rEl) have the same dimension (e.g., by Weyl’s Character Formula) and are therefore isomorphic. The result now follows for an arbitrary principal ideal domain R as in the proof of (3). By a partition we mean a sequence A = (A,, A,,...) of non-negative integers with I, > I, 2 .. . and A.,,,= 0 for all m large. If A, + 1= 0 we identify A with the partition (in the sense of [19, p. 51) (A,, A2,...,A,). For a partition 1 we denote by ‘A its , or conjugate ([ 19, p. 93). For each par- tition p = (pl, ,u2,...) there is defined an R-module L,(F) (the Schur functor for the partition p evaluated at F). We shall consider only the case in which 484 S. DONKIN

,~i dn (L,(F)=0 if p, >n). Since L,(F) is defined as the image of a GR- module map (see [ 11) it is a G,-module and hence (by the second paragraph of 2.6) also a G,(R) = G&(R)-module. The connection between the notation of [l-3] and ours is:

(5) For a partition 3, = (A,, &,..., A,) we have Y,(n)rL,,(F), as _C,- modules. Let ‘I = (pi ,..., Pi), By definition ([l, II 1.31) L,(F) is the image of a non-zero morphism

d,: A,F=A“‘FQ ... @A’+F+S,F=S,,(F)Q ... OS,“(F).

By (3) and (2) above A,F has a good filtration. Now A,F has a unique highest weight (si + . . . + Ed,)+ (si + . . . + E,,) + . . + (.si + .* . + .sp,),i.e., A,&, + A*&*+ ... + A,,&,,= 1 and 1 occurs with multiplicity one. Thus by Lemma 3.2.1 of [ 141 (in [ 143 the base ring is an algebraically closed field but the lemma is valid over a principal ideal domain and may be deduced from 2.6( 1) together with (1) above and the tensor identity for induction [S, (l.lO)]) there is a filtration O=No,N1,..., N,=A,F with NJN,- i E Y,(o,), 1 d i < 1, u, = 3, and oiUA for i-c 1. The module SAP also has a good filtration by (2) and (4). The multiplicity of Y,(r), r E A+ in a good filtration of SLF depends only on the formal character of SAP and so does not depend on which principal ideal domain R is. Thus to determine the multiplicity we may take the base ring to be an algebraically closed field of characteristic 0. Now combining from [19, the remark following 26.6 Theorem, 14.1 Young’s Rule and 4.13 Theorem] we obtain that Y,(n) occurs precisely once as a section in a good filtration of SAF and any occurring Y,(r) with t # 1 satisfies rD1. Thus, by Lemma 3.2.1 of [14], there is a filtration 0= V,,, Vi,..., V, = S,F with Vi/V,- i 2 Y,(zJ, TiEA+, 1 Qidm, such that t,=J and r,DA for i> 1. By [14, (1.5.3)] (which is valid over a principal ideal domain) we obtain Hom,,(N,/N,_ i, Vi/Vi_ i) = 0 if either 1 < i < 1 (for in this case ui < 2 < rj) or 1 < i < 1 and 1

special resolving system for GR. As in [ 161, we let A = A(n) be the R-sub- algebra of R[C,]= R@,Z[_G] generated by the cii= 10X, (l

A(n) = 6 A(n, r) r=O

(see [16, (2.lb)]), where A(n, r) is the set of in the cV which are homogeneous of total degree r, together with 0. We say that a polynomial _G,-module V is homogeneous of degree r if cf V) < A(n, r). Let V be a polynomial _G,-module. If Xi, X, are submodules of V which are polynomial of degree r then X, + X, is polynomial of degree r. It follows that there is a submodule, call it V(r), which is maximal among all those which are polynomial of degree r. We shall need: (6) V= cB,“=~V(r). This is due to Schur in the case R = C. The proof given in [ 16, p. 201 is for an infinite field but is equally valid, in our set-up, over a principal ideal domain. We can now prove: (7) A polynomial _G,-module has an Akin-Buchsbaum resolution if and only if it has a Schur filtration. By a Schur filtration we mean a filtration with each section either 0 or isomorphic to L,(F) for some partition p. Suppose V is a polynomial &- module which has an Akin-Buchsbaum resolution. Then V has a 6 resolution so that V has a good filtration by 2.6(4) and (1) above. Let Y,(n) be a section in a good filtration of V. Since V is polynomial, YR(IZ)is also polynomial (cf. [15, (1.2c)l). Also, Y,(n) is indecomposable and so, by (6), is polynomial of degree r for some r. It follows (cf. [ 16, (3.2c)l) that, for each weight 5 = (tl ,..., &J, all ti > 0 (and the sum of the C;iis r). In particular 1= (A, ,..., A,) E n + satisfies A, >O and YR(IZ)z L,,(F) by (5). Thus V has a Schur filtration. Now suppose that V has a Schur filtration. Suppose that V is a direct sum of submodules V’ and I”‘. Then V’ and V” have good filtrations by 486 S. DONKIN

Corollary 3.2.5 of [ 141. Moreover I” and V” are polynomial modules and so have Schur filtrations by the argument just given. If I” and I”’ each has an Akin-Buchsbaum resolution then, forming the direct sum, we obtain an Akin-Buchsbaum resolution of V. Thus we may suppose that V is indecomposable, in particular, by (5) that I/ is homogeneus of some degree r. By (1) above and 2.6(4), V has a resolution where each Ci belongs to 6.. From the definition of Cr;,it is clear that for any XEE, X@ (OyN) (where N=n”t;) is a direct sum of modules of the form /imlF@ ... 0 A’+F for all q sufficiently large. Thus tensoring the above sequence by @YN, for q sufficiently large, we obtain a resolution O+D,+Dmp,+ ... -+D,+ vQ(QqN)+o

where each Di is a direct sum of modules of the form described above. Since V has homogeneity degree r, V@ ( a4N) has degree r + qn and taking homogeneous components we obtain a resolution

O-+D,(r+qn)+D,P,(r+qn)+ ... -+D,(r+qn)+ V@(OqN)+O.

A summand AmIF@ ... @ AmSF of Di, with 0 < mj < n for all 1

O+D:,@K-+D:,-,@K+ ... +D&@K+VQK+O

where K = BqN and each Dj belongs to the Akin-Buchsbaum system. Ten- soring with the dual K* of K we obtain an Akin-Buchsbaum resolution of V. In particular we obtain from (7) the result of Akin-Buchsbaum [3]: (8) For each partition p, the Schur module L,(F) has an Akin-Buchsbaum resolution. It is also proved in [3] that the skew Schur module L,,JF), for par- titions 2, CL,has an Akin-Buchsbaum resolution. It is possible to set up, in a fairly general context, a theory of skew modules for reductive groups, show that a skew module has a good filtration and, by identifying these skew modules with those of Akin, Buchsbaum and Weyman, [l], in the case of GL,, obtain a proof of (8) with L,(F) replaced by L,,,(F). However, we postpone this until a later date. RESOLUTIONS OF MODULES FOR REDUCTIVE GROUPS 487

Finally, we mention that our theorem, on resolutions and liltrations, also applies to the modules WP#, p of James. Let p = (pl, pz,...) be a sequence of non-negative integers ,ui with finite sum, r say, and ,u# = (pL1#,/AT ,...) be a partition of s< r such that @

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